Properties

Label 68.1.f.a
Level 68
Weight 1
Character orbit 68.f
Analytic conductor 0.034
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
CM discriminant -4
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 68.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0339364208590\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.19652.1
Artin image $C_4\wr C_2$
Artin field Galois closure of 8.0.1257728.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + ( -1 - i ) q^{5} -i q^{8} + i q^{9} +O(q^{10})\) \( q + i q^{2} - q^{4} + ( -1 - i ) q^{5} -i q^{8} + i q^{9} + ( 1 - i ) q^{10} + q^{16} - q^{17} - q^{18} + ( 1 + i ) q^{20} + i q^{25} + ( 1 + i ) q^{29} + i q^{32} -i q^{34} -i q^{36} + ( -1 - i ) q^{37} + ( -1 + i ) q^{40} + ( 1 - i ) q^{41} + ( 1 - i ) q^{45} -i q^{49} - q^{50} + ( -1 + i ) q^{58} + ( -1 + i ) q^{61} - q^{64} + q^{68} + q^{72} + ( 1 + i ) q^{73} + ( 1 - i ) q^{74} + ( -1 - i ) q^{80} - q^{81} + ( 1 + i ) q^{82} + ( 1 + i ) q^{85} + ( 1 + i ) q^{90} + ( -1 - i ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{5} + 2q^{10} + 2q^{16} - 2q^{17} - 2q^{18} + 2q^{20} + 2q^{29} - 2q^{37} - 2q^{40} + 2q^{41} + 2q^{45} - 2q^{50} - 2q^{58} - 2q^{61} - 2q^{64} + 2q^{68} + 2q^{72} + 2q^{73} + 2q^{74} - 2q^{80} - 2q^{81} + 2q^{82} + 2q^{85} + 2q^{90} - 2q^{97} + 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/68\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(-1\) \(i\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
1.00000i
1.00000i
1.00000i 0 −1.00000 −1.00000 1.00000i 0 0 1.00000i 1.00000i 1.00000 1.00000i
55.1 1.00000i 0 −1.00000 −1.00000 + 1.00000i 0 0 1.00000i 1.00000i 1.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
17.c even 4 1 inner
68.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.1.f.a 2
3.b odd 2 1 612.1.l.a 2
4.b odd 2 1 CM 68.1.f.a 2
5.b even 2 1 1700.1.p.a 2
5.c odd 4 1 1700.1.n.a 2
5.c odd 4 1 1700.1.n.b 2
7.b odd 2 1 3332.1.m.b 2
7.c even 3 2 3332.1.bc.c 4
7.d odd 6 2 3332.1.bc.b 4
8.b even 2 1 1088.1.p.a 2
8.d odd 2 1 1088.1.p.a 2
12.b even 2 1 612.1.l.a 2
17.b even 2 1 1156.1.f.b 2
17.c even 4 1 inner 68.1.f.a 2
17.c even 4 1 1156.1.f.b 2
17.d even 8 2 1156.1.c.b 2
17.d even 8 2 1156.1.d.a 2
17.e odd 16 8 1156.1.g.b 8
20.d odd 2 1 1700.1.p.a 2
20.e even 4 1 1700.1.n.a 2
20.e even 4 1 1700.1.n.b 2
28.d even 2 1 3332.1.m.b 2
28.f even 6 2 3332.1.bc.b 4
28.g odd 6 2 3332.1.bc.c 4
51.f odd 4 1 612.1.l.a 2
68.d odd 2 1 1156.1.f.b 2
68.f odd 4 1 inner 68.1.f.a 2
68.f odd 4 1 1156.1.f.b 2
68.g odd 8 2 1156.1.c.b 2
68.g odd 8 2 1156.1.d.a 2
68.i even 16 8 1156.1.g.b 8
85.f odd 4 1 1700.1.n.b 2
85.i odd 4 1 1700.1.n.a 2
85.j even 4 1 1700.1.p.a 2
119.f odd 4 1 3332.1.m.b 2
119.m odd 12 2 3332.1.bc.b 4
119.n even 12 2 3332.1.bc.c 4
136.i even 4 1 1088.1.p.a 2
136.j odd 4 1 1088.1.p.a 2
204.l even 4 1 612.1.l.a 2
340.i even 4 1 1700.1.n.a 2
340.n odd 4 1 1700.1.p.a 2
340.s even 4 1 1700.1.n.b 2
476.k even 4 1 3332.1.m.b 2
476.z even 12 2 3332.1.bc.b 4
476.bb odd 12 2 3332.1.bc.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.f.a 2 1.a even 1 1 trivial
68.1.f.a 2 4.b odd 2 1 CM
68.1.f.a 2 17.c even 4 1 inner
68.1.f.a 2 68.f odd 4 1 inner
612.1.l.a 2 3.b odd 2 1
612.1.l.a 2 12.b even 2 1
612.1.l.a 2 51.f odd 4 1
612.1.l.a 2 204.l even 4 1
1088.1.p.a 2 8.b even 2 1
1088.1.p.a 2 8.d odd 2 1
1088.1.p.a 2 136.i even 4 1
1088.1.p.a 2 136.j odd 4 1
1156.1.c.b 2 17.d even 8 2
1156.1.c.b 2 68.g odd 8 2
1156.1.d.a 2 17.d even 8 2
1156.1.d.a 2 68.g odd 8 2
1156.1.f.b 2 17.b even 2 1
1156.1.f.b 2 17.c even 4 1
1156.1.f.b 2 68.d odd 2 1
1156.1.f.b 2 68.f odd 4 1
1156.1.g.b 8 17.e odd 16 8
1156.1.g.b 8 68.i even 16 8
1700.1.n.a 2 5.c odd 4 1
1700.1.n.a 2 20.e even 4 1
1700.1.n.a 2 85.i odd 4 1
1700.1.n.a 2 340.i even 4 1
1700.1.n.b 2 5.c odd 4 1
1700.1.n.b 2 20.e even 4 1
1700.1.n.b 2 85.f odd 4 1
1700.1.n.b 2 340.s even 4 1
1700.1.p.a 2 5.b even 2 1
1700.1.p.a 2 20.d odd 2 1
1700.1.p.a 2 85.j even 4 1
1700.1.p.a 2 340.n odd 4 1
3332.1.m.b 2 7.b odd 2 1
3332.1.m.b 2 28.d even 2 1
3332.1.m.b 2 119.f odd 4 1
3332.1.m.b 2 476.k even 4 1
3332.1.bc.b 4 7.d odd 6 2
3332.1.bc.b 4 28.f even 6 2
3332.1.bc.b 4 119.m odd 12 2
3332.1.bc.b 4 476.z even 12 2
3332.1.bc.c 4 7.c even 3 2
3332.1.bc.c 4 28.g odd 6 2
3332.1.bc.c 4 119.n even 12 2
3332.1.bc.c 4 476.bb odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(68, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{4} \)
$5$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
$7$ \( 1 + T^{4} \)
$11$ \( 1 + T^{4} \)
$13$ \( ( 1 + T^{2} )^{2} \)
$17$ \( ( 1 + T )^{2} \)
$19$ \( ( 1 + T^{2} )^{2} \)
$23$ \( 1 + T^{4} \)
$29$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
$31$ \( 1 + T^{4} \)
$37$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
$41$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
$43$ \( ( 1 + T^{2} )^{2} \)
$47$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$53$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$59$ \( ( 1 + T^{2} )^{2} \)
$61$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
$67$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$71$ \( 1 + T^{4} \)
$73$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
$79$ \( 1 + T^{4} \)
$83$ \( ( 1 + T^{2} )^{2} \)
$89$ \( ( 1 + T^{2} )^{2} \)
$97$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
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